Search results for: Gauss law

Authors: M. L. Burnwal, P. Raychowdhury

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Earthquake leads to adverse effects on buildings resting on soft soils. Mitigating the response of shallow foundations on soft soil with different methods reduces settlement and provides foundation stability. Few methods such as the rocking foundation (used in Performance-based design), deep foundation, prefabricated drain, grouting, and Vibro-compaction are used to control the pore pressure and enhance the strength of the loose soils. One of the problems with these methods is that the settlement is uncontrollable, leading to differential settlement of the footings, further leading to the collapse of buildings. The present study investigates the utility of geosynthetics as a potential improvement of the subsoil to reduce the earthquake-induced settlement of structures. A steel moment-resisting frame building resting on loose liquefiable dry soil, subjected to Uttarkashi 1991 and Chamba 1995 earthquakes, is used for the soil-structure interaction (SSI) analysis. The continuum model can simultaneously simulate structure, soil, interfaces, and geogrids in the OpenSees framework. Soil is modeled with PressureDependentMultiYield (PDMY) material models with Quad element that provides stress-strain at gauss points and is calibrated to predict the behavior of Ganga sand. The model analyzed with a tied degree of freedom contact reveals that the system responses align with the shake table experimental results. An attempt is made to study the responses of footing structure and geosynthetics with unreinforced and reinforced bases with varying parameters. The result shows that geogrid reinforces shallow foundation effectively reduces the settlement by 60%.

Keywords: settlement, shallow foundation, SSI, continuum FEM

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Authors: Mariem Saied, Jens Gustedt, Gilles Muller

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We present Dido, a source-to-source auto-generation and optimization framework for multi-dimensional stencil computations. It enables a large programmer community to easily and safely implement stencil codes on distributed-memory parallel architectures with Ordered Read-Write Locks (ORWL) as an execution and communication back-end. ORWL provides inter-task synchronization for data-oriented parallel and distributed computations. It has been proven to guarantee equity, liveness, and efficiency for a wide range of applications, particularly for iterative computations. Dido consists mainly of an implicitly parallel domain-specific language (DSL) implemented as a source-level transformer. It captures domain semantics at a high level of abstraction and generates parallel stencil code that leverages all ORWL features. The generated code is well-structured and lends itself to different possible optimizations. In this paper, we enhance Dido to handle both Jacobi and Gauss-Seidel grid traversals. We integrate temporal blocking to the Dido code generator in order to reduce the communication overhead and minimize data transfers. To increase data locality and improve intra-node data reuse, we coupled the code generation technique with the polyhedral parallelizer Pluto. The accuracy and portability of the generated code are guaranteed thanks to a parametrized solution. The combination of ORWL features, the code generation pattern and the suggested optimizations, make of Dido a powerful code generation framework for stencil computations in general, and for distributed-memory architectures in particular. We present a wide range of experiments over a number of stencil benchmarks.

Keywords: stencil computations, ordered read-write locks, domain-specific language, polyhedral model, experiments

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Authors: Thanida Sujarittham, Narumon Emarat, Jintawat Tanamatayarat, Kwan Arayathanitkul, Suchai Nopparatjamjomras

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Electric field is an important fundamental concept in electrostatics. In high-school, generally Thai students have already learned about definition of electric field, electric field due to a point charge, and superposition of electric fields due to multiple-point charges. Those are the prerequisite basic knowledge students holding before entrancing universities. In the first-year university level, students will be quickly revised those basic knowledge and will be then introduced to a more complicated topic—electric field due to continuous charged distributions. We initially found that our freshman students, who were from the Faculty of Science and enrolled in the introductory physic course (SCPY 158), often seriously struggled with the basic physics concepts—superposition of electric fields and inverse square law and mathematics being relevant to this topic. These also then resulted on students’ understanding of advanced topics within the course such as Gauss's law, electric potential difference, and capacitance. Therefore, it is very important to determine students' understanding of electric field due to continuous charged distributions. The open-ended question about sketching net electric field vectors from a uniformly charged insulating rod was administered to 260 freshman science students as pre- and post-tests. All of their responses were analyzed and classified into five levels of understandings. To get deep understanding of each level, 30 students were interviewed toward their individual responses. The pre-test result found was that about 90% of students had incorrect understanding. Even after completing the lectures, there were only 26.5% of them could provide correct responses. Up to 50% had confusions and irrelevant ideas. The result implies that teaching methods in Thai high schools may be problematic. In addition for our benefit, these students’ alternative conceptions identified could be used as a guideline for developing the instructional method currently used in the course especially for teaching electrostatics.

Keywords: alternative conceptions, electric field of continuous charged distributions, inverse square law, levels of student understandings, superposition principle

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Authors: K. Karuppasamy

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In this paper analysis of an infinite beam resting on multilayer tensionless extensible geosynthetic reinforced granular fill - poor soil system overlying soft soil strata under moving the load with constant velocity is presented. The beam is subjected to a concentrated load moving with constant velocity. The upper reinforced granular bed is modeled by a rough membrane embedded in Pasternak shear layer overlying a series of compressible nonlinear Winkler springs representing the underlying the very poor soil. The multilayer tensionless extensible geosynthetic layer has been assumed to deform such that at the interface the geosynthetic and the soil have some deformation. Nonlinear behavior of granular fill and the very poor soil has been considered in the analysis by means of hyperbolic constitutive relationships. Governing differential equations of the soil foundation system have been obtained and solved with the help of appropriate boundary conditions. The solution has been obtained by employing finite difference method by means of Gauss-Siedel iterative scheme. Detailed parametric study has been conducted to study the influence of various parameters on the response of soil – foundation system under consideration by means of deflection and bending moment in the beam and tension mobilized in the geosynthetic layer. These parameters include the magnitude of applied load, the velocity of the load, damping, the ultimate resistance of the poor soil and granular fill layer. The range of values of parameters has been considered as per Indian Railways conditions. This study clearly observed that the comparisons of multilayer tensionless extensible geosynthetic reinforcement with poor foundation soil and magnitude of applied load, relative compressibility of granular fill and ultimate resistance of poor soil has significant influence on the response of soil – foundation system. However, for the considered range of velocity, the response has been found to be insensitive towards velocity. The ultimate resistance of granular fill layer has also been found to have no significant influence on the response of the system.

Keywords: infinite beams, multilayer tensionless extensible geosynthetic, granular layer, moving load and nonlinear behavior of poor soil

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Authors: Kuheli Chakrabarty, Animesh Ghosh, Atanu Roy, Gourab Kanti Das

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Aldol reaction is an important class of carbon-carbon bond forming reactions. One of the popular ways to impose asymmetry in aldol reaction is the introduction of chiral auxiliary that binds the approaching reactants and create dissymmetry in the reaction environment, which finally evolves to enantiomeric excess in the aldol products. The last decade witnesses the usage of natural amino acids as chiral auxiliary to control the stereoselectivity in various carbon-carbon bond forming processes. In this context, L-proline was found to be an effective organocatalyst in asymmetric aldol additions. In last few decades the use of water as solvent or co-solvent in asymmetric organocatalytic reaction is increased sharply. Simple amino acids like L-proline does not catalyze asymmetric aldol reaction in aqueous medium not only that, In organic solvent medium high catalytic loading (~30 mol%) is required to achieve moderate to high asymmetric induction. In this context, huge efforts have been made to modify L-proline and 4-hydroxy-L-proline to prepare organocatalyst for aqueous medium asymmetric aldol reaction. Here, we report the result of our DFT calculations on asymmetric aldol reaction of benzaldehyde, p-NO2 benzaldehyde and t-butyraldehyde with a number of ketones using L-proline hydrazide as organocatalyst in wet solvent free condition. Gaussian 09 program package and Gauss View program were used for the present work. Geometry optimizations were performed using B3LYP hybrid functional and 6-31G(d,p) basis set. Transition structures were confirmed by hessian calculation and IRC calculation. As the reactions were carried out in solvent free condition, No solvent effect were studied theoretically. Present study has revealed for the first time, the direct involvement of two water molecules in the aldol transition structures. In the TS, the enamine and the aldehyde is connected through hydrogen bonding by the assistance of two intervening water molecules forming a supramolecular network. Formation of this type of supramolecular assembly is possible due to the presence of protonated -NH2 group in the L-proline hydrazide moiety, which is responsible for the favorable entropy contribution to the aldol reaction. It is also revealed from the present study that, water assisted TS is energetically more favorable than the TS without involving any water molecule. It can be concluded from this study that, insertion of polar group capable of hydrogen bond formation in the L-proline skeleton can lead to a favorable aldol reaction with significantly high enantiomeric excess in wet solvent free condition by reducing the activation barrier of this reaction.

Keywords: aldol reaction, DFT, organocatalysis, transition structure

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Authors: Nicoly Coelho, Eduardo Vieira Vilas Boas, Paulo Orestes Formigoni

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Faced with the remarkable developments in the various segments of modern engineering, provided by the increasing technological development, professionals of all educational areas need to overcome the difficulties generated due to the good understanding of those who are starting their academic journey. Aiming to overcome these difficulties, this article aims at an introduction to the basic study of numerical methods applied to fluid mechanics and thermodynamics, demonstrating the modeling and simulations with its substance, and a detailed explanation of the fundamental numerical solution for the use of finite difference method, using SCILAB, a free software easily accessible as it is free and can be used for any research center or university, anywhere, both in developed and developing countries. It is known that the Computational Fluid Dynamics (CFD) is a necessary tool for engineers and professionals who study fluid mechanics, however, the teaching of this area of knowledge in undergraduate programs faced some difficulties due to software costs and the degree of difficulty of mathematical problems involved in this way the matter is treated only in postgraduate courses. This work aims to bring the use of DFC low cost in teaching Transport Phenomena for graduation analyzing a small classic case of fundamental thermodynamics with Scilab® program. The study starts from the basic theory involving the equation the partial differential equation governing heat transfer problem, implies the need for mastery of students, discretization processes that include the basic principles of series expansion Taylor responsible for generating a system capable of convergence check equations using the concepts of Sassenfeld, finally coming to be solved by Gauss-Seidel method. In this work we demonstrated processes involving both simple problems solved manually, as well as the complex problems that required computer implementation, for which we use a small algorithm with less than 200 lines in Scilab® in heat transfer study of a heated plate in rectangular shape on four sides with different temperatures on either side, producing a two-dimensional transport with colored graphic simulation. With the spread of computer technology, numerous programs have emerged requiring great researcher programming skills. Thinking that this ability to program DFC is the main problem to be overcome, both by students and by researchers, we present in this article a hint of use of programs with less complex interface, thus enabling less difficulty in producing graphical modeling and simulation for DFC with an extension of the programming area of experience for undergraduates.

Keywords: numerical methods, finite difference method, heat transfer, Scilab

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Authors: Jiahui You, Kyung Jae Lee

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Hydraulic fracturing fluid (HFF) is widely used in shale reservoir productions. HFF contains diverse chemical additives, which result in the dissolution and precipitation of minerals through multiple chemical reactions. In this study, a new pore-scale Darcy–Brinkman–Stokes (DBS) model coupled with Level Set Method (LSM) is developed to address the microscopic phenomena occurring during the iron–HFF interaction, by numerically describing mass transport, chemical reactions, and pore structure evolution. The new model is developed based on OpenFOAM, which is an open-source platform for computational fluid dynamics. Here, the DBS momentum equation is used to solve for velocity by accounting for the fluid-solid mass transfer; an advection-diffusion equation is used to compute the distribution of injected HFF and iron. The reaction–induced pore evolution is captured by applying the LSM, where the solid-liquid interface is updated by solving the level set distance function and reinitialized to a signed distance function. Then, a smoothened Heaviside function gives a smoothed solid-liquid interface over a narrow band with a fixed thickness. The stated equations are discretized by the finite volume method, while the re-initialized equation is discretized by the central difference method. Gauss linear upwind scheme is used to solve the level set distance function, and the Pressure–Implicit with Splitting of Operators (PISO) method is used to solve the momentum equation. The numerical result is compared with 1–D analytical solution of fluid-solid interface for reaction-diffusion problems. Sensitivity analysis is conducted with various Damkohler number (DaII) and Peclet number (Pe). We categorize the Fe (III) precipitation into three patterns as a function of DaII and Pe: symmetrical smoothed growth, unsymmetrical growth, and dendritic growth. Pe and DaII significantly affect the location of precipitation, which is critical in determining the injection parameters of hydraulic fracturing. When DaII<1, the precipitation uniformly occurs on the solid surface both in upstream and downstream directions. When DaII>1, the precipitation mainly occurs on the solid surface in an upstream direction. When Pe>1, Fe (II) transported deeply into and precipitated inside the pores. When Pe<1, the precipitation of Fe (III) occurs mainly on the solid surface in an upstream direction, and they are easily precipitated inside the small pore structures. The porosity–permeability relationship is subsequently presented. This pore-scale model allows high confidence in the description of Fe (II) dissolution, transport, and Fe (III) precipitation. The model shows fast convergence and requires a low computational load. The results can provide reliable guidance for injecting HFF in shale reservoirs to avoid clogging and wellbore pollution. Understanding Fe (III) precipitation, and Fe (II) release and transport behaviors give rise to a highly efficient hydraulic fracture project.

Keywords: reactive-transport , Shale, Kerogen, precipitation

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Authors: Pulakesh Das, Mukunda Dev Behera, Manchiraju Sri Ramachandra Murthy

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Large numbers of people living in and around the Hindu Kush Himalaya (HKH) region, depends on this diverse mountainous region for ecosystem services. Following the global trend, this region also experiencing rapid population growth, and demand for timber and agriculture land. The eight countries sharing the HKH region have different forest resources utilization and conservation policies that exert varying forces in the forest ecosystem. This created a variable spatial as well altitudinal gradient in rate of deforestation and corresponding forest patch fragmentation. The quantitative relationship between fragmentation and demography has not been established before for HKH vis-à-vis along elevation gradient. This current study was carried out to attribute the overall and different nature in landscape fragmentations along the altitudinal gradient with the demography of each sharing countries. We have used the tree canopy cover data derived from Landsat data to analyze the deforestation and afforestation rate, and corresponding landscape fragmentation observed during 2000 – 2010. Area-weighted mean radius of gyration (AMN radius of gyration) was computed owing to its advantage as spatial indicator of fragmentation over non-spatial fragmentation indices. Using the subtraction method, the change in fragmentation was computed during 2000 – 2010. Using the tree canopy cover data as a surrogate of forest cover, highest forest loss was observed in Myanmar followed by China, India, Bangladesh, Nepal, Pakistan, Bhutan, and Afghanistan. However, the sequence of fragmentation was different after the maximum fragmentation observed in Myanmar followed by India, China, Bangladesh, and Bhutan; whereas increase in fragmentation was seen following the sequence of as Nepal, Pakistan, and Afghanistan. Using SRTM-derived DEM, we observed higher rate of fragmentation up to 2400m that corroborated with high human population for the year 2000 and 2010. To derive the nature of fragmentation along the altitudinal gradients, the Statistica software was used, where the user defined function was utilized for regression applying the Gauss-Newton estimation method with 50 iterations. We observed overall logarithmic decrease in fragmentation change (area-weighted mean radius of gyration), forest cover loss and population growth during 2000-2010 along the elevation gradient with very high R2 values (i.e., 0.889, 0.895, 0.944 respectively). The observed negative logarithmic function with the major contribution in the initial elevation gradients suggest to gap filling afforestation in the lower altitudes to enhance the forest patch connectivity. Our finding on the pattern of forest fragmentation and human population across the elevation gradient in HKH region will have policy level implication for different nations and would help in characterizing hotspots of change. Availability of free satellite derived data products on forest cover and DEM, grid-data on demography, and utility of geospatial tools helped in quick evaluation of the forest fragmentation vis-a-vis human impact pattern along the elevation gradient in HKH.

Keywords: area-weighted mean radius of gyration, fragmentation, human impact, tree canopy cover

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Authors: Riadh Zorgati, Thomas Triboulet

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In quite diverse application areas such as astronomy, medical imaging, geophysics or nondestructive evaluation, many problems related to calibration, fitting or estimation of a large number of input parameters of a model from a small amount of output noisy data, can be cast as inverse problems. Due to noisy data corruption, insufficient data and model errors, most inverse problems are ill-posed in a Hadamard sense, i.e. existence, uniqueness and stability of the solution are not guaranteed. A wide class of inverse problems in physics relates to the Fredholm equation of the first kind. The ill-posedness of such inverse problem results, after discretization, in a very ill-conditioned linear system of equations, the condition number of the associated matrix can typically range from 109 to 1018. This condition number plays the role of an amplifier of uncertainties on data during inversion and then, renders the inverse problem difficult to handle numerically. Similar problems appear in other areas such as numerical optimization when using interior points algorithms for solving linear programs leads to face ill-conditioned systems of linear equations. Devising efficient solution approaches for such system of equations is therefore of great practical interest. Efficient iterative algorithms are proposed for solving a system of linear equations. The approach is based on a preconditioning of the initial matrix of the system with an approximation of a generalized inverse leading to a stochastic preconditioned matrix. This approach, valid for non-negative matrices, is first extended to hermitian, semi-definite positive matrices and then generalized to any complex rectangular matrices. The main results obtained are as follows: 1) We are able to build a generalized inverse of any complex rectangular matrix which satisfies the convergence condition requested in iterative algorithms for solving a system of linear equations. This completes the (short) list of generalized inverse having this property, after Kaczmarz and Cimmino matrices. Theoretical results on both the characterization of the type of generalized inverse obtained and the convergence are derived. 2) Thanks to its properties, this matrix can be efficiently used in different solving schemes as Richardson-Tanabe or preconditioned conjugate gradients. 3) By using Lp norms, we propose generalized Kaczmarz’s type matrices. We also show how Cimmino's matrix can be considered as a particular case consisting in choosing the Euclidian norm in an asymmetrical structure. 4) Regarding numerical results obtained on some pathological well-known test-cases (Hilbert, Nakasaka, …), some of the proposed algorithms are empirically shown to be more efficient on ill-conditioned problems and more robust to error propagation than the known classical techniques we have tested (Gauss, Moore-Penrose inverse, minimum residue, conjugate gradients, Kaczmarz, Cimmino). We end on a very early prospective application of our approach based on stochastic matrices aiming at computing some parameters (such as the extreme values, the mean, the variance, …) of the solution of a linear system prior to its resolution. Such an approach, if it were to be efficient, would be a source of information on the solution of a system of linear equations.

Keywords: conditioning, generalized inverse, linear system, norms, stochastic matrix

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Authors: Ali Ateş, Ansar B. Mwimbo, Ali H. Abdulkarim

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General transport equation has a wide range of application in Fluid Mechanics and Heat Transfer problems. In this equation, generally when φ variable which represents a flow property is used to represent fluid velocity component, general transport equation turns into momentum equations or with its well known name Navier-Stokes equations. In these non-linear differential equations instead of seeking for analytic solutions, preferring numerical solutions is a more frequently used procedure. Finite difference method is a commonly used numerical solution method. In these equations using velocity and pressure gradients instead of stress tensors decreases the number of unknowns. Also, continuity equation, by integrating the system, number of equations is obtained as number of unknowns. In this situation, velocity and pressure components emerge as two important parameters. In the solution of differential equation system, velocities and pressures must be solved together. However, in the considered grid system, when pressure and velocity values are jointly solved for the same nodal points some problems confront us. To overcome this problem, using staggered grid system is a referred solution method. For the computerized solutions of the staggered grid system various algorithms were developed. From these, two most commonly used are SIMPLE and SIMPLER algorithms. In this study Navier-Stokes equations were numerically solved for Newtonian flow, whose mass or gravitational forces were neglected, for incompressible and laminar fluid, as a hydro dynamically fully developed region and in two dimensional cartesian coordinate system. Finite difference method was chosen as the solution method. This is a parametric study in which varying values of velocity components, pressure and Reynolds numbers were used. Differential equations were discritized using central difference and hybrid scheme. The discritized equation system was solved by Gauss-Siedel iteration method. SIMPLE and SIMPLER were used as solution algorithms. The obtained results, were compared for central difference and hybrid as discritization methods. Also, as solution algorithm, SIMPLE algorithm and SIMPLER algorithm were compared to each other. As a result, it was observed that hybrid discritization method gave better results over a larger area. Furthermore, as computer solution algorithm, besides some disadvantages, it can be said that SIMPLER algorithm is more practical and gave result in short time. For this study, a code was developed in DELPHI programming language. The values obtained in a computer program were converted into graphs and discussed. During sketching, the quality of the graph was increased by adding intermediate values to the obtained result values using Lagrange interpolation formula. For the solution of the system, number of grid and node was found as an estimated. At the same time, to indicate that the obtained results are satisfactory enough, by doing independent analysis from the grid (GCI analysis) for coarse, medium and fine grid system solution domain was obtained. It was observed that when graphs and program outputs were compared with similar studies highly satisfactory results were achieved.

Keywords: finite difference method, GCI analysis, numerical solution of the Navier-Stokes equations, SIMPLE and SIMPLER algoritms

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Authors: Fatna Benmessaoud, Mohammed Cheikh, Vencent Velay, Vanessa Vedal, Farhad Rezai-Aria, Christine Boher

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The global mechanical behavior of materials is strongly linked to their microstructure, especially their crystallographic texture and their grains morphology. These material aspects determine the mechanical fields character (heterogeneous or homogeneous), thus, they give to the global behavior a degree of anisotropy according the initial microstructure. For these reasons, the prediction of global behavior of materials in relationship with the microstructure must be performed with a multi-scale approach. Therefore, multi-scale modeling in the context of crystal plasticity is widely used. In this present contribution, a phenomenological elasto-viscoplastic model developed in the crystal plasticity context and finite element method are used to investigate the effects of crystallographic texture and grains sizes on global behavior of a polycrystalline equiaxed Ti-6Al-4V alloy. The constitutive equations of this model are written on local scale for each slip system within each grain while the strain and stress mechanical fields are investigated at the global scale via finite element scale transition. The beta phase of Ti-6Al-4V alloy modeled is negligible; its percent is less than 10%. Three families of slip systems of alpha phase are considered: basal and prismatic families with a burgers vector and pyramidal family with a burgers vector. The twinning mechanism of plastic strain is not observed in Ti-6Al-4V, therefore, it is not considered in the present modeling. Nine representative elementary volumes (REV) are generated with Voronoi tessellations. For each individual equiaxed grain, the own crystallographic orientation vis-à-vis the loading is taken into account. The meshing strategy is optimized in a way to eliminate the meshing effects and at the same time to allow calculating the individual grain size. The stress and strain fields are determined in each Gauss point of the mesh element. A post-treatment is used to calculate the local behavior (in each grain) and then by appropriate homogenization, the macroscopic behavior is calculated. The developed model is validated by comparing the numerical simulation results with an experimental data reported in the literature. It is observed that the present model is able to predict the global mechanical behavior of Ti-6Al-4V alloy and investigate the microstructural parameters' effects. According to the simulations performed on the generated volumes (REV), the macroscopic mechanical behavior of Ti-6Al-4V is strongly linked to the active slip systems family (prismatic, basal or pyramidal). The crystallographic texture determines which family of slip systems can be activated; therefore it gives to the plastic strain a heterogeneous character thus an anisotropic macroscopic mechanical behavior. The average grains size influences also the Ti-6Al-4V mechanical proprieties, especially the yield stress; by decreasing of the average grains size, the yield strength increases according to Hall-Petch relationship. The grains sizes' distribution gives to the strain fields considerable heterogeneity. By increasing grain sizes, the scattering in the localization of plastic strain is observed, thus, in certain areas the stress concentrations are stronger than other regions.

Keywords: microstructural parameters, multi-scale modeling, crystal plasticity, Ti-6Al-4V alloy

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Authors: Rubens P. Costa, Marcelo A. Leigui de Oliveira

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The comprehension of features on the energy spectra, the chemical compositions, and the origins of Ultra-High Energy Cosmic Rays (UHECRs) - mainly atomic nuclei with energies above ~1.0 EeV (exa-electron volts) - are intrinsically linked to the problem of determining the magnitude of their deflections in cosmic magnetic fields on cosmological scales. In addition, as they propagate from the source to the observer, modifications are expected in their original energy spectra, anisotropy, and the chemical compositions due to interactions with low energy photons and matter. This means that any consistent interpretation of the nature and origin of UHECRs has to include the detailed knowledge of their propagation in a three-dimensional environment, taking into account the magnetic deflections and energy losses. The parameter space range for the magnetic fields in the universe is very large because the field strength and especially their orientation have big uncertainties. Particularly, the strength and morphology of the Extragalactic Magnetic Fields (EGMFs) remain largely unknown, because of the intrinsic difficulty of observing them. Monte Carlo simulations of charged particles traveling through a simulated magnetized universe is the straightforward way to study the influence of extragalactic magnetic fields on UHECRs propagation. However, this brings two major difficulties: an accurate numerical modeling of charged particles diffusion in magnetic fields, and an accurate numerical modeling of the magnetized Universe. Since magnetic fields do not cause energy losses, it is important to impose that the particle tracking method conserve the particle’s total energy and that the energy changes are results of the interactions with background photons only. Hence, special attention should be paid to computational effects. Additionally, because of the number of particles necessary to obtain a relevant statistical sample, the particle tracking method must be computationally efficient. In this work, we present an analysis of the propagation of ultra-high energy charged particles in the intergalactic medium. The EGMFs are considered to be coherent within cells of 1 Mpc (mega parsec) diameter, wherein they have uniform intensities of 1 nG (nano Gauss). Moreover, each cell has its field orientation randomly chosen, and a border region is defined such that at distances beyond 95% of the cell radius from the cell center smooth transitions have been applied in order to avoid discontinuities. The smooth transitions are simulated by weighting the magnetic field orientation by the particle's distance to the two nearby cells. The energy losses have been treated in the continuous approximation parameterizing the mean energy loss per unit path length by the energy loss length. We have shown, for a particle with the typical energy of interest the integration method performance in the relative error of Larmor radius, without energy losses and the relative error of energy. Additionally, we plotted the distance amplification from rectilinear propagation as a function of the traveled distance, particle's magnetic rigidity, without energy losses, and particle's energy, with energy losses, to study the influence of particle's species on these calculations. The results clearly show when it is necessary to use a full three-dimensional simulation.

Keywords: cosmic rays propagation, extragalactic magnetic fields, magnetic deflections, ultra-high energy

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